1. What is the twist of TMDs? Como, June 12, 2013 Oleg Teryaev JINR, Dubna

2. Outline Definitions of twist TMDs as infinite towers of twists Quarks in vacuum and inside the hadrons: TMDs vs non-local condensates HT resummation and analyticity in DIS HT resummation and scaling variables: DIS vs SIDIS

3. HT resummation in DIS Higher Twists in spin-dependent DIS: GDH sum rule – finite sum of infinite number of divergent terms Resummation of HT and analyticity Comparing modified scaling variables

4. What is twist? Power corrections ~1/Q 2 ------//------------- ~ M 2 DIS – it’s the same (~ M 2 /Q 2 ) TMD – usually ~1/Q 2 - (M 2 /k T 2 ) i attributed to Leading Twist However – tracing the powers of M is helpful for studying HT in coordinate (~impact parameter) space

5. Collins FF and twist 3 x(T) –space : qq correlator ~ M - twist 3 Cf to momentum space (k T /M ) – M in denominator – “LT” x k T spaces Moment – twist 3 (for Sivers – Boer, Mulders, Pijlman) Higher (2D-> Bessel) moments – infinite tower of twists (for Sivers - Ratcliffe,OT)

6. Resummation in x-space (DY) Full x/k T – dependence DY weighted cross-section Similarity with non-local quark condensate: quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?! –cf with Radyushkin et al Universal hadron(type-dependent)/vacuum functions?!

7. Hadronic vs vacuum matrix elements Hadron-> (LC) momentum; dimension-> twist; quark virtuality -> TM; (Euclidian) space separation -> impact parameter D-term ~ Cosmological constant in vacuum; Negative D-> negative CC in space-like/positive in time-like regions: Annihilation~Inflation!

8. Spin dependent DIS Two invariant tensors Only the one proportional to contributes for transverse (appears in Born approximation of PT) Both contribute for longitudinal Apperance of only for longitudinal case –result of the definition for coefficients to match the helicity formalism

9. Generalized GDH sum rule Define the integral – scales asymptotically as At real photon limit (elastic contribution subtracted) – - Gerasimov-Drell-Hearn SR Proton- dramatic sign change at low Q 2 !

10. Finite limit of infinite sum of inverse powers?! How to sum c i (- M 2 /Q 2 ) i ?! May be compared to standard twist 2 factorization Light cone: Lorentz invariance Summed by representing

11. Summation and analyticity? Justification (in addition to nice parton picture) - analyticity! Correct analytic properties of virtual Compton amlitude Defines the region of x Require: Analyticity of first moment in Q 2 Strictly speaking – another integration variable (Robaschik et al, Solovtsov et al)

12. Summation and analyticity! Parton model with |x| < 1 – transforms poles to cuts! – justifies the representation in terms of moments For HT series c i = - moments of HT “density”- geometric series rather than exponent: Σ c i (- M 2 /Q 2 ) = Like in parton model: pole -> cut Analytic properties  proper integration region (positive x, two-pion threshold) Finite value for Q 2 =0: - - inverse moment!

13. Summation and analyticity “Chiral” expansion: - (- Q 2 /M 2 ) i “Duality” of chiral and HT expansions: analyticity allows for EITHER positive OR negative powers (no complete series!) Analyticity – (typically)alternating series Analyticity of HT  analyticity of pQCD series – (F)APT Finite linit -> series starts from 1/Q 2 unless the density oscillates Annihilation – (unitarity - no oscillations) justification of “short strings”?

14. Short strings Confinement term in the heavy quarks potential – dimension 2 (GI OPE – 4!) scale ~ tachyonic gluon mass Effective modification of gluon propagator

15. Decomposition of (J. Soffer, OT ‘92) Supported by the fact that Linear in, quadratic term from Natural candidate for NP, like QCD SR analysis – hope to get low energy theorem via WI (C.f. pion F.F. – Radyushkin) - smooth model For -strong Q – dependence due to Burkhardt-Cottingham SR

16. Models for :proton Simplest - linear extrapolation – PREDICTION (10 years prior to the data) of low (0.2 GeV) crossing point Accurate JLAB data – require model account for PQCD/HT correction – matching of chiral and HT expansion HT – values predicted from QCD SR (Balitsky, Braun, Kolesnichenko) Rather close to the data For Proton

17. The model for transition to small Q (Soffer, OT ’04)

18. Models for :neutron and deuteron Access to the neutron – via the (p-n) difference – linear in -> Deuteron – refining the model eliminates the structures for neutron and deuteron

19. Duality for GDH – resonance approach Textbook (Ioffe, Lipatov. Khoze) explanation of proton GGDH structure – contribution of dominant magnetic transition form factor Is it compatible with explanation?! Yes!– magnetic transition contributes entirely to and as a result to

20. Bjorken Sum Rule – most clean test Strongly differs from smooth interpolation for g 1 (Ioffe,Lipatov,Khoze) Scaling down to 1 GeV

21. New option: Analytic Perturbation Theory Shirkov, Solovtsov: Effective coupling – analytic in Q 2 Generic processes: FAPT (BMS) Does not include full NPQCD dynamics (appears at ~ 1GeV where coupling is still small) –> Higher Twist Depend on (A)PT Low Q – very accurate data from JLAB

22. Bjorken Sum Rule-APT Accurate data + IR stable coupling -> low Q region

23. PT/HT duality

24. Matching in PT and APT Duality of Q and 1/Q expansions

25. 4-loop corrections included V.L. Khandramai, R.S. Pasechnik, D.V. Shirkov, O.P. Solovtsova, O.V. Teryaev. Jun 2011. 6 pp. e-Print: arXiv:1106.6352 [hep-ph] V.L. KhandramaiR.S. PasechnikD.V. ShirkovO.P. SolovtsovaO.V. Teryaev HT decrease with PT order and becomes compatible to zero (V.I. Zakharov’s duality) Analog for TMD – intrinsic/extrinsic TM duality!?

26. Asymptotic series and HT Duality: HT can be eliminated at all (?!) May reappear for asymptotic series - the contribution which cannot be described by series due to its asymptotic nature.

27. Another version of IR stable coupling – “gluon mass” – Cornwall,.. Simonov,.. Shirkov(NLO) arXiv:1208.2103v2 [hep-th] 23 Nov 2012 HT – in the “VDM” form M 2 /(M 2 + Q 2 ) Corresponds to f(x) ~ Possible in principle to go to arbitrarily small Q BUT NO matching with GDH achieved Too large average slope – signal for transverse polarization (cf Ioffe e.a. interpolation)!

28. Account for transverse polarization -> descripyion in the whole Q region (Khandramai, OT, in progress) 1-st order – LO coupling with (P) gluon mass + (NP) “VDM” GDH – relation between P and NP masses

29. NP vs P masses Non-monotonic! “Phase diagram”

30. P/NP masses

31. Data at LO

32. NLO

33. Data vs NLO

34. Modification of spectral function for HT Add const × Q 2 /(M 2 2+Q 2 ) 2 -> First of second derivative of delta-function appear – double and triple poles (single – almost cancelled) Masses: P= 0.68 NP=0.76 Expansion at low Q 2 Real scale – pion mass?!

35. HT – modifications of scaling variables (L-T relations) Various options since Nachtmann ~ Gluon mass -//- new (spectrality respecting) modification JLD representation

36. Resummed twists: Q->0 (D. Kotlorz, OT)

37. Modified scaling variable for TMD First appeared in P. Zavada model X Z = Suggestion – also (partial) HT resummation(M goes from denominator to numerator in cordinate/impact parameter space)?!

38. Conclusions/Discussion TMD – infinite towers of twists Similar to non-local quark condensates – vacuum/hadrons universality?! Infinite sums of twists – important for DIS at Q->0 Representation for HT similar to parton model: preserves analyticity changing the poles to cuts Modified scaling variables – models for twists towers at DIS and (TMD) SIDIS Good description of the data at all Q 2 with the single scale parameter